or the moment of inertia about any line through 0, is measured by the square of the reciprocal of the radius vector of this ellipsoid which coincides with the line.

This is called the momental ellipsoid. It has no physical existence, but is an artifice to bring under the methods of geometry the properties of moments of inertia. The momental ellipsoid has a definite form for every point of a rigid body.

If this ellipsoid be referred to another set of axes, and its equation become

a'§3 + b'n2 + c'§2 — 2ď′n5 — 2e'§§ — 2ƒ'§n = 1,

the coefficients a', b', c' will be the moments of inertia about the new axes, and d, e, f' will be the products.

Now every ellipsoid has three axes, to which if it is referred its equation takes the form

Ağ2 + Bn2 + C¿2 = 1.

With respect to these axes, the products of inertia vanish.

8. Hence we see that the moment of inertia about one of the principal axes is the greatest, and about another the least possible. It was from this property that Euler, who first thoroughly investigated the subject, gave them the

name.

It is now clear, that for all questions depending only on moments and products of inertia, any body may be replaced by its momental ellipsoid. And farther, that any two systems which have the same momental ellipsoid at a point, are about that point kinetically identical.

If the moments of inertia of a body about three axes at right angles through a point are equal, the ellipsoid becomes a sphere. They are therefore equal about all axes, and every axis is a principal axis. The body is then said to be kinetically symmetrical with respect to that point. Thus a cube is kinetically symmetrical about its centre.

The following question is of some interest.

9. Under what circumstances is there a point in a body such that the moments of inertia about all axes through it are equal?

If there is such a point, all sets of axes through it are principal axes.

Let the co-ordinates of the point referred to the principal axes at the centre of mass be a, b, c. Then the products of inertia of the body about the parallel axes through the point are

m.bc, m.ca, m.ab,

for those about the axes through the centre of mass are

zero.

If all axes at the point are to be principal axes, these must be so;

.. bc=0, ca=0, ab=0,

equations which require that two of a, b, c should be zero.

Let b=0, c = 0, then the point required lies on the axis of x,-one of the set of principal axes at the centre of mass.

But further, it is necessary that the moments of inertia about these axes should be equal. Let A, B, C be the moments of inertia about the axes through the centre of mass. Then those about the parallel axes through the point required are

A, B+ma2, C+ ma2.

If these are to be equal, we have

BC and A-B= ma3.

Hence our condition is, that two of the principal moments at the centre of mass should be equal. In other words, the momental ellipsoid at the centre of mass must be a spheroid. And then the point lies on the unequal axis at a distance from the centre equal to

80

EXAMPLES.

1. Given A, B the moments of inertia of a body about two principal axes Ox, Oy, prove that the product of inertia about axes Ox, Oy in the same plane, inclined to the former set at an angle a,

2. Prove that any two of the principal moments of inertia are together greater than the third.

3. No ellipsoid except a sphere can be its own momental ellipsoid at its centre.

4. Every elliptic plate is similar to the section of its momental ellipsoid made by its own plane.

5. If a, b, c are the semiaxes of the momental ellipsoid of a rigid body in order of magnitude, shew that

6. Given the angular velocity of a body which is rotating about a fixed point; about what axis must it be rotating, so that the angular momentum shall be greatest?

7. Two systems of equal mass have the same principal axes, and the same moments of inertia about them at some one point; prove that they have the same principal axes at any point, and the same moments of inertia about any axis.

8. Prove that there can always be found three points, one on each of the three principal axes of any system at any point, such that the moments and products of inertia of

three suitable equal masses collected at them, are equal to the moments and products of inertia of the system about any axes whatever through that point.

9. If these equal masses be each one-third of the mass of the system (m), shew that the distances of the three points along the axes are

10. In a triangular plate ABC, D is the middle point of BC, and E the foot of the perpendicular let fall from A on BC. Shew that the middle point of DE is the point at which BC is a principal axis.

11. Shew that the difference of the moments of inertia of a body round two axes in a given plane which are equally inclined to a fixed line in the same plane, is proportional to the sine of the angle between those axes.

P. G.

6

IX.

CASES OF MOTION WITHOUT ROTATION.

1. THE complete solution of a problem of motion would involve the finding of the position of the system at a given time, of the velocities at a given time or in a given position, and of the values of any previously unknown forces, such as pressures or frictions which may act on the system. If the forces are impulsive, only velocities and forces can be required; for the position is unaltered during the impulse, and to follow the subsequent changes belongs to a separate problem of the other kind. Questions of impulsive motion can then always be solved, for the changes of velocity and the forces appear as unknown quantities in equations which are in general simple algebraical equations. But if the forces are of the kind called finite, the equations of motion are differential equations of the second order as regards co-ordinates of position. In some simple cases these can be completely solved and the requirements of the above solution satisfied; but in more complicated cases we can get no farther than a first integral, that is, an algebraical equation giving the velocities. In such a case to find the position at a given time is impossible. Our demands must be limited by what we can get; and the words "to find the motion" have come to mean, "to find the velocities of the system in any position.

We will therefore in general use velocities and their first differential coefficients in the expressions for the effective forces; but if, in any case, the co-ordinates of position must enter, we can use their first and second differential coefficients to express velocities and accelerations.